Computing the Minimal Perimeter Polygon for Sets of Rectangular Tiles based on Visibility Cones

Abstract

To study convexity properties of digital planar objects, the minimum perimeter polygon (MPP) was defined in the 1970 s in articles by Sklansky, Chazin, Hansen, Kibler, and Kim, where pixels were identified with polygonal tiles in mosaics, and two algorithms (1972, 1976) were proposed to determine the MPP vertices. These algorithms are based on constructing and iteratively restricting visibility cones, the MPP vertices result as special vertices of the tiles. The present paper proposes a novel MPP algorithm for objects given as regular complexes in rectangular mosaics, which are edge-adjacency-connected sets of tiles that have neither end tiles nor holes and whose boundaries not necessarily are simple. The new algorithm takes as input the canonical boundary path, we also propose a boundary tracing algorithm to obtain this path. We review the two classic MPP algorithms for rectangular tiles and a simplified adaptation for square tiles that is recommended in widely used modern textbooks on digital image analysis (2018, 2020) to produce approximations of simple digital 4-contours. We show that all these algorithms fail and that their mathematical basis is flawed, we correct the errors to develop the new MPP algorithm. Our MPP algorithm is illustrated using examples and its correctness is proved. Under our assumptions, the MPP coincides with the relative convex hull of a set A with respect to a polygon \(B\supset A\), where A is not necessarily a polygon, not even connected.